Problem 17 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{m}{n}-1}{\frac{m}{n}+1} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{m-n}{m+n}\).

Step 1: Rewrite the Expression

Start by rewriting the complex fraction as follows:\[\frac{\frac{m}{n} - 1}{\frac{m}{n} + 1} = \frac{\frac{m}{n} - \frac{n}{n}}{\frac{m}{n} + \frac{n}{n}}\]This helps clear the terms for the next step, where we need a common denominator.

Step 2: Combine Fractions in the Numerator and Denominator

Combine the fractions in both the numerator and the denominator to get a single fraction:Numerator: \(\frac{m}{n} - \frac{n}{n} = \frac{m - n}{n}\)Denominator: \(\frac{m}{n} + \frac{n}{n} = \frac{m + n}{n}\)So, the complex fraction becomes:\[\frac{\frac{m-n}{n}}{\frac{m+n}{n}}\]

Step 3: Simplify by Dividing Fractions

To simplify further, divide the two fractions:\[\frac{\frac{m-n}{n}}{\frac{m+n}{n}} = \frac{m-n}{n} \times \frac{n}{m+n}\]This simplifies, as the \(n\) in the numerator and denominator cancels out:\[\frac{m-n}{m+n}\]

Step 4: Final Simplification

Since there are no further terms to factor or simplify in the expression \(\frac{m-n}{m+n}\), this is the simplest form of the original complex fraction.

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form so that they are easier to understand and work with. In the case of complex fractions, simplifying means removing the layers of divisions within the fraction and breaking it down to a standard, single-layer fraction.
To simplify, follow these steps:

• Convert complex fractions to a simple fraction by dividing the numerator by the denominator. This typically involves fraction division.
• Cancel out common factors in the numerator and the denominator, if any, to get it to its simplest form.
• Ensure the fraction expresses the lowest possible whole numbers as parts.

This approach not only makes the expressions easier to manage but also helps in identifying patterns or solutions more swiftly.

Numerator and Denominator

Every fraction has two integral components: the numerator and the denominator. The numerator is the number on top and represents the part of the whole.
While the denominator is at the bottom and it represents the total number of equal parts the whole is divided into.

• In a simple fraction \(rac{a}{b}\), \(a\) is the numerator and \(b\) is the denominator.
• In complex fractions, these can themselves be fractions; for instance, \(\frac{\frac{a}{b} - \frac{c}{d}}{\frac{e}{f} + \frac{g}{h}}\).
• The process involves separating the boundaries clearly by resolving expressions both in the numerator and the denominator independently before merging them.

Having a clear understanding of these terms helps in applying arithmetic operations more effectively and simplifies the processing of complex operations.

Common Denominator

Obtaining a common denominator is crucial when dealing with complex fractions. It allows you to combine fractions with different denominators into a single, cohesive fraction.
A common denominator is typically the least common multiple of the individual denominators in a complex fraction.
Here's how you find it:

• Identify the denominators you want to combine.
• Calculate the least common multiple (LCM) of these denominators.
• Adjust each fraction so that all fractions have this common denominator for easy addition or subtraction.

By doing this, the equation becomes easier to solve, and the transition from complex fractions to simplified forms becomes smoother.

Fraction Division

Fraction division is fundamental to simplifying complex fractions. It involves dividing one fraction by another, typically where one is the numerator and the other the denominator.

• First, rewrite the division into a multiplication problem by flipping the second fraction (known as the reciprocal).
• For example, to divide \(\frac{a}{b}\) by \(\frac{c}{d}\), multiply \(\frac{a}{b}\) by \(\frac{d}{c}\).
• Perform the multiplication straightforwardly: multiply the numerators together and the denominators together.

Remember, multiplying with reciprocals is the same as dividing in fractions, simplifying the process and leading down to a single, simpler fraction in most cases.